We are tasked with finding the probability of the sum being 7, given that one of the dice shows a 3. This is a conditional probability problem. We will break it down using the following steps:

Step 1: Determine the Sample Space Given the Condition

When one die shows a 3, the other die can show any of the values from 1 to 6. So, the possible outcomes where one die shows a 3 are:

• If the first die is 3, the second die can be any of 1, 2, 3, 4, 5, or 6.
• If the second die is 3, the first die can be any of 1, 2, 3, 4, 5, or 6.

Therefore, the total number of outcomes where at least one die is showing a 3 is:

• Outcomes when the first die is 3: (3,1), (3,2), (3,3), (3,4), (3,5), (3,6) — 6 outcomes.
• Outcomes when the second die is 3: (1,3), (2,3), (3,3), (4,3), (5,3), (6,3) — 6 outcomes.

However, the outcome (3,3) is counted twice, so the total number of unique outcomes is: $6 + 6 - 1 = 11 \text{ outcomes.}$

Step 2: Find the Outcomes Where the Sum is 7

Now, we need to identify the outcomes where the sum of the dice is 7, and one of the dice shows a 3. These outcomes are:

• (3,4)
• (4,3)

Thus, there are 2 favorable outcomes where the sum is 7 and at least one die is showing a 3.

Step 3: Calculate the Conditional Probability

The conditional probability $P(\text{Sum} = 7 \mid \text{One die is 3})$ is given by the formula: $P(\text{Sum} = 7 \mid \text{One die is 3}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes given the condition}}.$ From the previous steps, we know that there are 2 favorable outcomes and 11 possible outcomes where one die is 3. Thus, the conditional probability is: $P(\text{Sum} = 7 \mid \text{One die is 3}) = \frac{2}{11}.$

Final Answer:

The probability of getting a sum of 7, given that one of the dice shows a 3, is $\frac{2}{11}$.

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Would you like more details or have any further questions?

Related questions:

1. What is the probability of rolling a sum of 7 with two dice without any conditions?
2. How can conditional probability be applied in real-world situations?
3. What is the probability of getting a sum of 7, given that the sum is greater than 5?
4. What is the probability of rolling a sum of 7 or 11 in a single throw of two dice?
5. How does the concept of conditional probability extend to more than two dice?

Tip:

For problems involving conditional probability, always carefully define the sample space under the given condition, as this simplifies the problem and ensures accuracy.